Frobenius-Schur indicators for some fusion categories associated to symmetric and alternating groups

TL;DR

This paper computes Frobenius-Schur indicators for fusion categories derived from inclusions of symmetric, alternating, and cyclic groups, extending previous results and providing new insights into their structure.

Contribution

It generalizes earlier findings on Frobenius-Schur indicators for bismash product Hopf algebras to broader fusion categories, offering new proofs and settling open questions.

Findings

Calculated indicator values for specific group inclusions.

Extended results to fusion categories beyond Hopf algebras.

Provided shorter, more general proofs.

Abstract

We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusion categories not associated to Hopf algebras.